Question: Simplify the following expression and state the condition under which the simplification is valid. You can assume that $t \neq 0$. $z = \dfrac{-8}{5t + 7} \div \dfrac{5t}{3t(5t + 7)} $
Explanation: Dividing by an expression is the same as multiplying by its inverse. $z = \dfrac{-8}{5t + 7} \times \dfrac{3t(5t + 7)}{5t} $ When multiplying fractions, we multiply the numerators and the denominators. $z = \dfrac{ -8 \times 3t(5t + 7) } { (5t + 7) \times 5t } $ $ z = \dfrac {-8 \times 3t(5t + 7)} {5t (5t + 7)} $ $ z = \dfrac{-24t(5t + 7)}{5t(5t + 7)} $ We can cancel the $5t + 7$ so long as $5t + 7 \neq 0$ Therefore $t \neq -\dfrac{7}{5}$ $z = \dfrac{-24t \cancel{(5t + 7})}{5t \cancel{(5t + 7)}} = -\dfrac{24t}{5t} = -\dfrac{24}{5} $